Right, I saw these mentioned on the internet, and read up a bit about them, they're an alternate algebraic closure of the rational field.

So, after my maths teacher, an obsessive pure-maths sort I'll call T, mentions he'll be doing infinities in the next class (I'm in the top set of double maths, so he can afford to waste lessons on fun stuff like that). He also runs maths club, which I don't normally go to, because it'd mean walking back to college after I've gone home, and I'm lazy.

But, I thought it might be worth mentioning to him the concept of surreal numbers, inside of which can be constructed not only the real numbers, but infinite numbers, and infinitesimal numbers. He patiently listens to me go over the concept, and then says "Cool." Then I suggest that it would be a good concept to cover in maths club. He then says that's a good idea, and would I do the talk on surreal numbers, as I am the only one in college who understands them.

Now, my normal reaction when faced with the prospect of talking to a large group of people is to bolt for the door, but I decided that it might be worth it this time, what with the whole gaining the respect of my fellow nerds thing. So I agreed, and I just need to get my notes in order for the talk, and then give them a few weeks notice to publicise (!) it. Not only that, but my mum, a psychology teacher, and her friend, a sociology teacher, are going to show up. That ought to be fun.

So, that's my nervous involvement with them, and as well as making this post not completely useless for non-maths types, explains why I'm discussing such an esoteric subject all of a sudden.

Now for the maths:

This subject makes heavy use of set notation, and I have tried to make this as painless as possible, but I'm warning you not to continue if you have a headache.

There are two important definitions in the field of surreal numbers, the two you absolutely must have to get started.

Definition of a surreal number(Note: S is the set of surreal numbers, no blackboard bold, sorry. and the ∋ sign is the wrong way round, but it would be even more confusing to write these out backwards, so sorry, just keep on your mental toes)

{L|R} ∋ S ⇔ ∀l∋L:l∋S ^ ∀r∋R:r∋S ^ ∀l∋L∀r∋R:¬(r≤l)

Definiton of less than or equal to(note: Xl means the left set of the surreal number x, similarly with Xr)

x≤y ⇔ ¬∃m ∋ Xl:y≤m ^ ¬∃n ∋ Yr:n ≤ x

Now, these definitions may seem circular, but that's because they are. Mathematicians prefer "iterative", however.

Now, the simplest set possible is the empty set, {}, which is also represented by the symbol ∅.

So, let us consider the set {∅|∅}. Is this a surreal number? Well, first let us consider the statement

∀x∋∅:x∋S

Is it true? Yes, trivially so. Any condition is true for all members of the empty set. That covers the first and second part of the surreal number definition. Now let us look at the third:∀x∋∅∀y∋∅:¬(y≤x)Again, trivially true, as there are no members in the empty set.

So, {∅|∅} is a surreal number, and to save time, I shall assign it the label 0. At the moment these labels are arbitrary, but when it comes to mapping the reals as a subset of the surreals, they will be meaningful (and it can be done, the reals are a proper subset of the surreals).

The first is false, as 0≤0. Therefore, the statement {{0}|∅}≤{∅|∅} is false, and by our conventions, we can say 0<{{0}|∅}. So, let us label this number 1. Lo! We hath achieved much. From nothing, and structure, we have created unity.

So it shall be called 1/2. This is a good label, because when surreal addition is defined, {{0}|{1}}+{{0}|{1}}=1.

Anyway, now {{-1}|{1}}. {{-1}|{1}}=0; that is to say, {{-1}|{1}}≤0 ^ 0≤{{-1}|{1}}.

That's a significantly surprising result that it justifies proving. So, in academic tradition, I'll leave it as an exercise to the reader.

Anyway, you probably get the idea that we can represent all integers in surreal number form, and 1/2. But we can do 1/8, and 1/16 too. Now, these statements would take a definition of multiplicative inverse over the surreals to justify, but let me just state the mapping for all fractions m/2n:f(x)={f(m-1/2n)|f(m+1/2n)}

Now, one more definition: addition. Addition is the basic operation of arithmetic, and all others can be derived from it. Over the surreals it is defined as:

x+y={Xl+y, x+Yl|Xr+y, x+Yr}

Again, this is circular. But, as any operation done to the null set results in the null set, it is usable. Take 1+2,

{{0}|∅}+{{1}|∅}={{0}+{{1}|∅}, {{0}|∅}+{1}|∅+{{1}|∅}, {{0}|∅}+∅}

Which simplifies to:{{0}+{{1}|∅}, {{0}|∅}+{1}|∅}

Which leaves two sums,0+{{1}|∅}={∅+{{1}|∅}, {0}+{1}|∅+{{1}|∅}, 0+∅}={{0}+{1}|∅}{{0}|∅}+1={{0}+{1}, {1}+∅|∅+{1},{1}+∅}={{0}+{1}|∅}Which leaves us with one sum needed to do, 0+1.{∅|∅}+{{0}|∅}={∅+{1}, {0}+{0}|∅+{1},{0}+∅}={{0}+{0}|∅}

now, 0+0:{∅|∅}+{∅|∅}={∅+{0}, {0}+∅|∅+{0}, {0}+∅}={∅|∅}=0

now, substituting back in:0+1={{0}|∅}=1

0+2={{1}|∅}=21+1={{1}|∅}=2

1+2={{0}+{2}, {1}+{1}|∅}={{2},{2}|∅}

Which, dues to a property of surreal numbers, is equal to {{2},∅}. A sensible name for which, from our mapping, is 3.

Yes, I know it's a lot simpler to calculate in real arithmetic, but that's not the point. Anyway, S is abelian over +. And the number that was called '-1' really is the additive inverse of 1. Subtraction is merely shorthand for adding negative numbers.And multiplication is shorthand for repeated addition. So, essentially, they are all defined. Yes, this is a sloppy job, but it's good enough for an introductory text.

Now, a few rules for the surreals (all have been proved, but I wouldn't want to spoil your fun by depriving you of mathematical exercise):

The simplification theorem: you can remove all but the biggest of L and the smallest of R without changing the value of a surreal number.

A surreal number is greater than all members of its left set and lesser than all members of its right set.

A surreal number is equal to the oldest number between the largest member of its left set, and the smallest member of its right set. (oldest means the one that you would have to go through the least number of generative iterations to get to)

Anyway, this could all be done in the reals to far, so let me show you something interesting:

Consider the set Zs, the set of surreal integers. It is defined thusly:

0∋Zsn∋Zs⇒{n|∅}∋Zsn∋Zs⇒{∅|n}∋Zs

As you can probably tell, this is identical to the set Z of integers. So, I'll call it Z for simplicity's sake. Anyway, consider the number: {Z|∅}. And, yes, it is a number. It meets all the requirements. But what is its value?

Well, the set Z contains all numbers that can be created in the form 1+1+1+... Therefore, by the second bulletpoint, it is greater than them. So therefore, it must be infinity. Oh, I'm sure you've had your maths teacher tell you infinity is not a number. But that's in the reals. But, we don't call it infinity. Infinity is too vague. The proper name for this number is ω. It is an ordinal. All ordinal infinities can be constructed in the surreals. You can calculate ω-1, ω+ω, ω2, ωω, etc. And the smallest ordinal not constructible using ω, by using the set of all ordinals constructible from ω in the left set. It is called ω1, and the smallest ordinal not constructible using ω1 is called ω2. Neat, eh? You could make up a complete set of ordinal arithmetic.

But also, consider the set D of unit dyadic fractions:

1∋Dn∋D ⇒ 1/2n ∋D

i.e., {1, 1/2, 1/4, 1/8...}

Now, consider the number {∅|D}. What value does it have? Well, it is smaller than all the unit dyadic fractions. But whatever positive fraction you pick, there will be a member of D that is smaller. So, this number is smaller than all possible fractions of a unit. But, it is bigger than zero. This is an infinitesimal number, which I shall call ε. Even though ε is smaller than all possible fractions, you can still have ε/2. You can have 2ε, 1+ε, and ωε (which is equal to one). This creates another whole section of arithmetic to play with.

And this also leads to a real, genuinely interesting exercise for the reader: How do you represent a non-dyadic fraction as a surreal number? Of course, like all my 'exercises for the reader', you could look this up on Google. That's why this has so many holes in, it's just an introduction. But, if you think you've followed what I've explained so far, working this one out might actually be worth your time.

Neat, eh?

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Quote from: The Will

In a movie, one can always pull back and condemn the character or the artist ... But in playing a game ... we can be encouraged to examine our own values by seeing how we behave within virtual space.

After seeing you post that post-modernist generated essay, I'm going have to say your full of yourself. You just love to have people comment on how wonderfully smart you are. Don't worry, I'm sure someone here will feed your ego the size of downtown Manhatten.

Seriously, going from "I discovered dividing by 0!" to "surreal numbers" is just funny, if anything. I'll have to pull a quote from "3001: A Space Odyssey" and say: "There was a mathematician that used something called surreal mathematics to prove that there are an infinite number of grades between deist and theist. He went insane."

I'll see if this is anything but utter male cow manure once I go to a computer that supports the type style. From a precursory, half-info glance I think you try to use false math proofs (i.e. 1=2) and obscure jargon to render the reader dazed and in awe, much like the post-modernist essay generator you stumbled upon.

But it may be legit. Since I don't support unicode, I can only guess, but if it's anything it's group theory. And to give a lecture about something/publish a paper, shouldn't you formulate a couple of solid proofs of your own?

After seeing you post that post-modernist generated essay, I'm going have to say your full of yourself. You just love to have people comment on how wonderfully smart you are. Don't worry, I'm sure someone here will feed your ego the size of downtown Manhatten.

Seriously, going from "I discovered dividing by 0!" to "surreal numbers" is just funny, if anything.

Those were jokes. This is not.

EDIT: Also, Manhattan.

One thing to consider: this contained not only a backstory, but was much longer than the joke posts. I don't put that much effort into jokes, I'm lazy.

Quote

From a precursory, half-info glance I think you try to use false math proofs (i.e. 1=2) and obscure jargon to render the reader dazed and in awe, much like the post-modernist essay generator you stumbled upon.

Except that this contains no logical fallacies. Unlike post-modernism, in mathematics, there is no way to string together meaningful symbols in a syntactically correct way, and come up with, as you put it, "utter male cow manure".

Quote

But it may be legit. Since I don't support unicode, I can only guess, but if it's anything it's group theory.

Oh, how nice of you. Here's a tip: if you can't read something, don't run off at the mouth at it.

And, it's set theory. I said, in non-Unicode characters, "they're an alternate algebraic closure of the rational field"

Oh, and actually, here's another tip: If you think I'm making stuff up, Google it. I even say to at the end of the post. (Well, close as).

There is no excuse for making yourself look a pillock because you can't be bothered to type two words into Google. That's lazier than me.

Quote

And to give a lecture about something/publish a paper, shouldn't you formulate a couple of solid proofs of your own?

It's maths club, not a meeting of maths PhDs. I'm just introducing the concept to them.

...anyone who wants to understand, read through until you get stuck. Then ask me. I need plenty of question-answering practice.

If it was bull, then I wouldn't bother saying that, would I? I'd just enjoy people getting stuck and thinking I'm so smart. But I'm not. I want to explain things. But instead of asking me for an explanation, you made a pronouncement about a post you couldn't read, without referring to any outside sources. Infact, I'd go so far as to say that you are full of yourself.

« Last Edit: January 13, 2008, 08:00:12 pm by Quantum Burrito »

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Quote from: The Will

In a movie, one can always pull back and condemn the character or the artist ... But in playing a game ... we can be encouraged to examine our own values by seeing how we behave within virtual space.

Asked my mathematician dad. He knows nothing about "surreal" numbers. Is there a different term for them?

Oh, and could you explain in sentences? Writing out stuff in math language looks smart, but others would understand more if you actually go through it in layman terms.

EDIT: Nevermind. He just uses a different term. I looked it up on wikipedia.

What is the point you are trying to make with this thread? "I am smart?" Surreal numbers are unimportant, and have no value in the real world. They are abstract mathematical concepts that pure mathematicians like to eat for breakfast, unlike real and imaginary numbers (Used in designing circuitry), which actually have importance. It is an esoteric mathematical concept.

I found an odd way that any untrue statement can be turned into a true statement.

3=43 x 0 = 4 x 00=00 + 3 = 0 + 33=3

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Asked my mathematician dad. He knows nothing about "surreal" numbers. Is there a different term for them?

Oh, and could you explain in sentences? Writing out stuff in math language looks smart, but others would understand more if you actually go through it in layman terms.

No other term for them in my knowledge. But as I said, a quick Google search will make it pretty evident they exist. Your dad might want to read On Numbers And Games by John Conway, if he's never heard of them.

And the post was ridiculously long in compact mathematical notation. But I'll try translating the beginning.

First off, a set is just a collection of objects. {1,2,3} is a set, as is {apple, orange, green}. Continuing sets are written ending in an ellipsis, i.e., N={1,2,3,4,5...} for N, the set of natural (counting) numbers.

Definition of a surreal number(Note: S is the set containing all surreal numbers, the letter should be in a font called "blackboard bold", but I can't to that on the forums. Also, the ∋ sign is the wrong way round, but it would be even more confusing to write these out backwards, so sorry, just keep on your mental toes)

This means:Given a set containing two other sets, a left set 'L' and a right set 'R', that set is a surreal number if and only if all the objects in the sets L and R are surreal numbers, and no member of the set R is less than or equal to any member of the set L.

Definition of less than or equal toWhen constructing a new number system, you have to forget everything you know about numbers. This includes things like 1=1, x+0=x, etc. all must be built from first principles. The only comparison operator actually defined is the 'less than or equal to operator', and while you may have an intuitive sense of what it does, disregard that. This is the definition used in the surreal numbers, and although it shall end up acting pretty much intuitively, you must construct things with this definition.

Here it is in mathspeak:(note: Xl means the left set of the surreal number x, similarly with Xr)x≤y ⇔ ¬∃m ∋ Xl:y≤m ^ ¬∃n ∋ Yr:n ≤ x

This means:a surreal number x is less than or equal to a surreal number y if and only if y is less than or equal to no member of x's left set (remember all surreal numbers have the form {L|R}, with L being the 'left set' and R being the 'right set'). And no member of y's right set is less than or equal to x.

Now, this definition may seem circular, but that's because it is. Mathematicians prefer you call it "iterative", however.

So, how can this work? Well, we are saved from an infinite loop by the properties of the "empty set", which does exactly what it says on the tin. It looks like this: {}, and is represented by the symbol ∅. Now, the importan properties are this:

Anything is true for all members of the empty set. It is true that all members of the empty set are greater than zero, and also that all members of the empty set are less than zero. This is because there are no members in the empty set.

There does not exist anything the the empty set. This means that if there is a statement beggining "there exists in the empty set", it is automatically false, and if a statement starts "there does not exist in the empty set", it is automatically true.

These are important. This is how you can create something from nothing. Ask your dad if you think this sounds wrong, he'll definitely tell you it is correct.

Now, consider the surreal number {∅|∅}. That is, the number who's left and right sets contain no elements. Now, there are two requirements for this to actually be a surreal number. One: "all the objects in the left set and the right set are surreal numbers". This is true by the first property of the empty set that I stated. The second is "no member of the right set is less than or equal to any member of the left set". Well the right set is the empty set, and so that statement is true by the second property of the empty set I stated.

So, it meets all two of the requirements for being a surreal number. Therefore, it is one. one of the things you do in maths with a useful set like {∅|∅} (useful because it is the only surreal number we have at the moment) is give it a label. Seeing as it contains nothing, 0 is an appropriate label. So we shall say that, in the surreal numbers, 0={∅|∅}, by definition.

Right, that's the start translated. Tell me if you understand it or not, so I can refine it, and eventually move on to the next section.

I found an odd way that any untrue statement can be turned into a true statement.

3=43 x 0 = 4 x 00=00 + 3 = 0 + 33=3

Please tell me this is a joke?

EDIT: Oops, jumping to conclusions. TTT, your statements have no relevance in mathematics, as they start with an untrue statement. It doesn't matter that the end is true, try to turn it around to get a falsehood from truth, and you'll end up dividing by zero.

« Last Edit: January 13, 2008, 08:53:31 pm by Quantum Burrito »

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Quote from: The Will

In a movie, one can always pull back and condemn the character or the artist ... But in playing a game ... we can be encouraged to examine our own values by seeing how we behave within virtual space.